Method and device for selecting optimal transform matrices for down-sampling DCT image

ABSTRACT

Down-sampling of an image may be performed in the DCT domain. Transform matrices are obtained for down-sampling a DCT image of size M×N to a down-sampled DCT image of size I×J. The transform matrices may be used to down-sample the DCT image directly in the DCT domain. A spatial domain down-sampling method is selected and applied to the DCT image to produce a down-sampled DCT reference image. The transform matrices are selected by solving an optimization problem, leading to transform matrices which achieve a desired trade-off between the visual quality of images obtained using the transform matrices and the computational complexity associated with using the transform matrices. The visual quality is a measure of the difference between the down-sampled DCT image obtained using the transform matrices and the visual quality of the DCT reference image obtained using a spatial domain down-sampling method.

This application is a continuation of U.S. patent application Ser. Nos.13/155,584, filed Jun. 8, 2011 and 12/133,095, filed Jun. 4, 2008 whichclaims the benefit of U.S. Provisional Application No. 60/941,794, filedJun. 4, 2007. Each patent application identified above is incorporatedherein by reference.

FIELD OF THE INVENTION

The present invention relates to image manipulation and moreparticularly to the down-sampling of an image to obtain an image havinga smaller size.

BACKGROUND OF THE INVENTION

As wireless and wired network connectivity is rapidly expanding and thenumber of network users is steadily increasing, there has been greatmomentum in the multimedia industry for supporting content display in awide variety of network connected devices. This is complicated, however,due to the great diversity of devices with various display resolutions.The range of devices includes both standard and high-definition computermonitors with large displays, and smart phone devices with smallerdisplays. Accordingly, it is often necessary to down-sample an image toa size which may be displayed on a given device. Down-sampling may alsobe desirable in order to minimize the bandwidth or time required totransfer an image over a network. This is particularly a concern wherethe image is being transferred over a wireless medium.

Many image and video types represent images or video in a DiscreteCosine Transform (“DCT”) domain. For example the JPEG, MPEG, DV andMJPEG image and video types represent images or videos in the DCTdomain. Where the image or video is represented in a DCT domain,down-sampling is particularly difficult.

Typically, down-sampling a DCT image or video involves first convertingDCT data to a spatial domain using an inverse DCT operation and applyinga standard down-sampling method in the spatial domain. After the imageis down-sampled in the spatial domain it is typically converted back tothe DCT domain using a DCT operation. Due to the use of the DCT and theinverse DCT, methods which down-sample a DCT image or video by firstconverting it back to the spatial domain typically have a largecomputational complexity. A large computational complexity results in alonger calculation and a calculation which utilizes more processor timeand power.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference will now be made, by way of example, to the accompanyingdrawings which show an embodiment of the present application, and inwhich:

FIG. 1 shows a method of obtaining transforming matrices fordown-sampling a DCT image according to one aspect of the presentapplication;

FIG. 2 shows a method of obtaining a training set in accordance with oneaspect of the present application;

FIG. 3 shows a method of selecting transform matrices in accordance withone aspect of the present application;

FIGS. 4A and 4B show a three layer network representation of thetransform matrices;

FIG. 5 shows a method of training transform matrices using a learningwith forgetting algorithm;

FIG. 6 shows a method of training transform matrices using a selectivelearning with forgetting algorithm;

FIG. 7 shows a method of obtaining a down-sampled DCT image inaccordance with one aspect of the present application; and

FIG. 8 shows an arrangement of electrical devices to be used in carryingout methods of the present application.

Similar reference numerals are used in different figures to denotesimilar components.

DESCRIPTION OF SPECIFIC EMBODIMENTS

In one embodiment of the present application, a method is provided fordown-sampling a DCT image of size M×N directly in a DCT domain byapplying two transform matrices. The first matrix, called a rowtransform matrix, D_(IM), is used to down-sample the number of rows inthe DCT image. That is, it may be used to convert a DCT image, C_(MN),with M rows into a DCT matrix with I rows. The number of columns in therow transform matrix is equal to the number of rows in the DCT image.The second matrix, called a column transform matrix, W_(NJ), is used todown-sample the number of columns in the DCT image. That is, it may beused to convert a DCT image with N columns into a DCT matrix with Jcolumns. The number of rows in the column transform matrix is equal tothe number of columns in the DCT image. Accordingly, the row transformmatrix is of a size I×M and the column transform matrix is of size N×J.

The DCT image, C_(MN), may be any DCT image including, for example, aJPEG image. In some instances the DCT image may represent a portion of avideo, such as a frame of a video. For example, the DCT image may be asingle frame of a video of the DV video type.

A down-sampled DCT image is obtained using the transform matrices bymultiplying the row transform matrix, D_(IM), with the product of theDCT image, C_(MN), and the column transform matrix, W_(NJ). Thedown-sampled DCT image may be obtained using a transformation formula:D _(IM) ·C _(MN) ·W _(NJ)

Since one of the transform matrices is applied to the DCT image on eachside of the DCT image, the transform matrices may be called lineartransform double sided matrices (LTDS). That is, one of the transformmatrices is applied to the left of the DCT image in the transformationformula and one is applied to the right of the DCT image in thetransformation formula. In this case, the row transform matrix isapplied to the left of the DCT image and the column transform matrix isapplied to the right of the DCT image.

A wide range of spatial-domain down-sampling methods may be representedas a LTDS in the DCT domain. For example, in one embodiment, thetransform matrices may be determined as:D _(IM) =T _(II,t) ·E _(IM) ·A _(MM) *·L _(MM) ·A _(MM) ·T _(MM,t′), andW _(NJ) =T _(NN,t) ·B _(NN) ·R _(NN) ·B _(NN) *·G _(NJ) ·T _(JJ,t′),where T_(SS) is a matrix of size S×S and which is equal to:

${T_{{SS},t} = \begin{pmatrix}\lbrack t\rbrack & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \lbrack t\rbrack\end{pmatrix}},$where t is a DCT matrix.

E_(IM) and G_(NJ) are linear interpolation matrices for use indown-sampling in the spatial domain, and A_(MM) is an M×M DFT transformmatrix with its element given by

${a_{uv} = {\frac{1}{\sqrt{M}}{\exp\left( \frac{{- j}\; 2\pi\;{uv}}{M} \right)}}},$u=0, 1, . . . , M−1, v=0, 1, . . . , M−1 and A_(MM)* is its conjugate.Similarly, B_(NN) is an N×N DFT transform matrix and B_(NN)* is itsconjugate matrix. L_(MM) and R_(NN) are diagonal matrixes with diagonalelements being L_(M1) and R_(1N) respectively. L_(M1) and R_(1N) are twoone-dimensional filters.

The LTDS representation above may be obtained by concatenating theinverse DCT, spatial domain down-sampling of a certain format, and theDCT. More specifically, a spatial domain image X_(MN) may be obtained byapplying an inverse DCT to a DCT image C_(MN): X_(MN)=t′□C_(MN)□t, wheret is a DCT matrix, t′ is a transpose of the DCT matrix and □ denotesblock-wise multiplications. Then, a spatial-domain method may beselected and used to down-sample X_(MN) to obtain an I×J image, denotedas x_(IJ). Finally, the DCT may be applied to the I×J image, resultingin V_(IJ)=t□x_(IJ)□t′, where V_(IJ) is a down-sampled DCT image.

Generally, down-sampling in the spatial domain consists of two steps:low-pass filtering and interpolation. In some embodiments, a low passfilter may be implemented based on a 2D discrete Fourier transform(DFT). Given an image X_(MN), the filtering output {tilde over (X)}_(MN)is obtained by:{tilde over (X)} _(MN) =A _(MM)*·((A _(MM) ·X _(MN) ·B _(NN))

F _(MN))·B_(MN)*,where A_(MM) is an M×M DFT transform matrix with its element given by

${a_{uv} = {\frac{1}{\sqrt{M}}{\exp\left( \frac{{- j}\; 2\pi\;{uv}}{M} \right)}}},$u=0, 1, . . . , M−1, v=0, 1, . . . , M−1 and A_(MM)* is its conjugate.Similarly, B_(NN) is an N×N DFT transform matrix and B_(NN)* is itsconjugate matrix. F_(MN) is the low-pass filtering matrix in the DFTdomain and the symbol

denotes element wise multiplications.

If F_(MN) is constructed using two one-dimensional filters (i.e.F_(MN)=L_(M1)·R_(IN)) then the element wise multiplication may beremoved, yielding {tilde over(X)}_(MN)=A_(MM)*·L_(MM)·(A_(MM)·X_(MN)·B_(NN))·R_(NN)·B_(NN)*, whereL_(MM) and R_(NN) are diagonal matrixes with diagonal elements beingL_(M1) and R_(1N) respectively.

A linear interpolation filter may then be applied. The linearinterpolation filter to be applied is in the form of interpolationmatrices, E_(1M), G_(NJ). Therefore, the spatial domain down-sampledimage may be computed asx _(IJ) =E _(1M) ·{tilde over (X)} _(MN) ·G _(NJ) =E _(1M) ·A _(MM) *·L_(MM) ·A _(MM) ·X _(MN) ·B _(NN) ·R _(NN) ·B _(NN) *·G _(NJ)

The concatenation of inverse DCT, spatial domain down sampling and DCTis therefore:V _(IJ) =t□[E _(1M) ·A _(MM) *·L _(MM) ·A _(MM)·(t′□C _(MN) □t)·B _(NN)·R _(NN) ·B _(NN) *·G _(NJ) ]□t′

The block wise multiplication can then be replaced by applying a resultoft□C _(MN) =T _(MM,t) ·C _(MN),where

$T_{{MM},t} = \begin{pmatrix}\lbrack t\rbrack & 0 & 0 \\0 & \ddots & 0 \\0 & 0 & \lbrack t\rbrack\end{pmatrix}$and t is a DCT matrix.

Consequently, a linear transform in the DCT domain can be obtained as:V _(IJ) =D _(IM) ·C _(MN) ·W _(NJ) where D _(IM) =T _(II,t) ·E _(IM) ·A_(MM) *·L _(MM) ·A _(MM) ·T _(MM,t′), and W _(NJ) =T _(NN,t) ·B _(NN) ·R_(NN) ·B _(NN) *·G _(NJ) ·T _(JJ,t′).

That is, the row transform matrix may be represented as:D_(IM)=T_(II,t)·E_(IM)·A_(MM)*·L_(MM)·A_(MM)·T_(MM,t′). Similarly, thecolumn transform matrix may be represented asW_(NJ)=T_(NN,t)·B_(NN)·R_(NN)·B_(NN)*·G_(NJ)·T_(JJ,t′). The rowtransform matrix and the column transform matrix may be obtained bydetermining a solution to the equations:D_(IM)=T_(II,t)·E_(IM)·A_(MM)*·L_(MM)·A_(MM)·T_(MM,t′) andW_(NJ)=T_(NN,t)·B_(NN)·R_(NN)·B_(NN)*·G_(NJ)·T_(JJ,t′)

Accordingly, in one aspect of the present application a method ofdown-sampling a DCT image is provided. In one embodiment, the methodincludes steps of determining a row transform matrix using the equation:D_(IM)=T_(II,t)·E_(IM)·A_(MM)*·L_(MM)·A_(MM)·T_(MM,t′) and determining acolumn transform matrix using the equation:W_(NJ)=T_(NN,t)·B_(NN)·R_(NN)·B_(NN)*·G_(NJ)·T_(JJ,t′)

As illustrated in FIG. 1, in another embodiment of the presentapplication, a method is provided for obtaining transform matrices fordown-sampling a DCT image of size M×N to a down-sampled DCT image ofsize I×J directly in the DCT domain.

The method of FIG. 1 will permit a row transform matrix D_(IM) and acolumn transform matrix W_(NJ) to be selected which jointly maximize thevisual quality of down-sampled DCT images that are obtained by applyingthe transform matrices to the DCT images while minimizing thecomputational complexity of applying the transform matrices to the DCTimages to obtain the down-sampled DCT images. That is, a row transformmatrix and a column transform matrix are selected which provide adesired trade-off between computational complexity and visual quality.It is typically considered desirable to have a higher qualitydown-sampled DCT image and a lower computational complexity associatedwith obtaining that down-sampled DCT image.

At step 102 optimization parameters may be obtained. The optimizationparameters permit a user, system administrator, or programmer to havecontrol over the balance between computational complexity and visualquality. For example, the optimization parameters may include acomputational complexity trade-off parameter, ρ, which is used tobalance the relative importance of various factors on the computationalcomplexity. In some embodiments, the optimization parameters may includea rounding parameter, η, which is used to determine how close a valuemust be to another value before the algorithm will attempt to round tothe other value. The optimization parameters may also include a qualityand complexity trade-off parameter λ which is used to adjust the desiredtrade-off between the visual quality and complexity. That is, thequality and complexity trade-off parameter λ may be used to specifywhether optimization of the transform matrices will be more or lessfocused on visual quality or complexity. The optimization parameters mayalso include selective learning with forgetting threshold parameters,w_(o), d_(o,), which will be used in a selective learning withforgetting stage to adjust the threshold above which an element in thetransform matrices will be considered protected and not pushed towardszero. In some embodiments, the optimization parameters may include alearning step size parameter, α, which is used to adjust the step sizeof learning operations. That is, it is used to adjust how great thedegree with which the transform matrices will be adjusted after aniteration of learning.

Depending on the method used to optimize the transform matrices, some orall of the optimization parameters discussed above may be obtained instep 102. In some embodiments, these parameters are stored in a memoryand they are retrieved from the memory.

In other embodiments, the optimization parameters may be obtained from auser. A user may be prompted to enter desired optimization parametersusing an input device. In some embodiments, a setup module may be run bythe user which allows the user to enter optimization parameters. Afterthe user enters the optimization parameters, they may be saved to thememory.

In some embodiments, the optimization parameters are entered on a serverwhich obtains the transform matrices.

Next, at step 104, a training set is obtained which will be used intraining the transform matrices. The training set includes at least oneDCT reference image, C_(MN), and typically includes a correspondingdown-sampled DCT reference image, V_(IJ), for each reference image. TheDCT reference image is of the size M×N and the down-sampled DCTreference image is of the size I×J.

As will be described more fully below, the training set will be used toevaluate the quality of down-sampled images obtained using the transformmatrices relative to the quality of the down-sampled DCT referenceimage.

Typically, the training set will consist of more than one DCT referenceimage C_(MN) and more than one down-sampled DCT reference image, V_(IJ).The down-sampled DCT reference image V_(IJ) is obtained using anotherdown-sampling method. Typically, the down-sampled DCT reference imageV_(IJ) is obtained using a spatial domain reference image in which theDCT reference image C_(MN) is converted to the spatial domain using aninverse DCT transform, and the spatial domain image is down-sampledusing a spatial domain down-sampling method. Then, the down-sampledspatial domain reference image is transferred back to the DCT domainusing a DCT transform to obtain the down-sampled DCT reference image,V_(IJ).

This will permit the transform matrices to be trained multiple times,thus increasing the accuracy of the transform matrices. For example, insome embodiments the training set includes five DCT reference imagesC_(MN) and five down sampled DCT reference images, V_(IJ), eachcorresponding to one of the DCT reference images.

The step 104 of obtaining a training set may include a step of applyinga pre-selected down sampling method to a DCT reference image to obtainthe down-sampled DCT reference images. Typically, the pre-selecteddown-sampling method is a spatial domain down-sampling method.

Referring now to FIG. 2, a method is shown for obtaining a training setusing a spatial domain down-sampling method. At step 202 a down-samplingmethod is selected. Typically, the selected down-sampling method will bea down-sampling method which produces a down-sampled DCT reference imagewith an optimal visual quality since the visual quality of thedown-sampled image produced according to the present method is dependenton the visual quality of the down-sampled DCT reference image.

In some instances, the down-sampling method is pre-determined.

Where the down-sampling method is a spatial domain down-sampling method,the method of obtaining the training set includes a step 204 ofconverting at least one of the DCT reference images to the spatialdomain by applying an inverse DCT to produce a spatial domain referenceimage. Next, at step 206, the spatial domain reference image isdown-sampled using the selected spatial domain down-sampling method toproduce a spatial domain down-sampled reference image. Then, at step208, the spatial domain down-sampled reference image is converted backto the DCT domain to obtain the down-sampled DCT reference image.

Typically, the step 206 of down-sampling the converted DCT referenceimage in the spatial domain includes the steps of: 1) applying a lowpass filter to the spatial domain reference image to produce a filteredimage; and 2) interpolating within the filtered image to produce thespatial domain down-sampled reference image.

With respect to the step of filtering, a filter is typically selectedwhich provides a good trade-off among three factors: the reduction orelimination of aliasing, the preservation of low-frequency components,and the reduction or elimination of ringing. Generally, a filter with asharp transition band provides a good performance on anti-aliasing andpreserving low-frequency components, yet a sharp transition band incursringing along intensity edges in the filtered image.

In some embodiments, a Butterworth filter is used. The Butterworthfilter provides a good trade-off among the three factors (aliasing,low-frequency components, and ringing). That is, two one dimensionalButterworth filters may be selected with the frequency responsefunction:

${{H(f)}} = \sqrt{\frac{1}{1 + \left( {f/f_{c}} \right)^{2L}}}$where f_(c) is the cutoff frequency and L represents the order of thefilter which characterizes the transition band. In some embodiments, thefilter may be a tenth order filter (i.e. L=10).

As noted above, this filter may be applied to the spatial domainreference image to obtain a filtered image.

Following the step of low-pass filtering, interpolation may be performedon the filtered image to obtain the spatial domain down-sampledreference image. Interpolation involves estimating the values of thespatial domain down-sampled reference image at certain points.

A wide range of interpolation methods may be used. For example, nearestneighbour interpolation or spline interpolation may be used. TheNyquist-Shannon interpolation uses a sinc function which achievestheoretically optimal performance.

In some embodiments, cubic interpolation may be used which provides asmooth function with a continuous first derivative. While the nearestneighbor interpolation corresponds to a square interpolation function,the bicubic interpolation is obtained by convolving three squarefunctions.

It will be apparent to a person skilled in the art that, while specificspatial domain down-sampling methods have been discussed above, themethod discussed herein may be used with any other spatial domainmethod.

The training set may be stored in a memory after it is created. In someembodiments, the memory may contain a database of training sets forvarious down-sampling ratio. For example, the database may contain datatraining sets to train transform matrices to convert a matrix of sizeM×N to a matrix of any one of a number of various sizes (i.e. I×J, O×P,Q×R, etc. . . . ).

In embodiments where training sets are previously generated and storedin memory, the step 104 of obtaining a training set may merely requirethe training set to be retrieved from the memory.

Once a training set is obtained, the method of obtaining transformmatrices for down-sampling a DCT image includes a step 106 of selectingtransform matrices that minimize an optimization problem. Theoptimization problem is a function of the error between the down-sampledimage obtained by applying the transform matrices to the reference imageand the down-sampled DCT reference image. The optimization problem isalso a function of the computational complexity of applying thetransform matrices.

It will be appreciated that various methods may be employed to findtransform matrices which jointly optimize the visual quality andcomputational complexity. In some embodiments, discussed more fullybelow, a multi-layer network may be used to find transform matriceswhich solve the optimization problem.

Accordingly, in step 106, transform matrices are found which obtain asuitable balance between maximizing the quality of the down-sampledimage obtained using the matrices, while minimizing the computationalcomplexity associated with down-sampling using the transform matrices.That is, an LTDS is found with the best trade-off between the fidelityof a down-sampled image and the computational complexity of the usingthe LTDS in the sense of minimizing the joint cost.

In order to select the transform matrices which minimize the joint cost,a value representing the visual quality of the down-sampled DCT imagesobtained using the transform matrices may be determined. A valuerepresenting the computational complexity associated with applying thetransform matrices to a DCT image in order to down-sample the DCT imagemay also be determined. The joint cost of the transform matrices maythen be determined as a function of the value representing the visualquality of the down-sampled DCT images obtained using the transformmatrices and the computational complexity associated with applying thetransform matrices to a DCT image in order to down-sample the DCT image.

The visual quality of the down-sampled image is measured relative to thequality of the down-sampled DCT reference image obtained using the otherdown-sampling method. Where quality is measured in this manner, thelinear transform double sided matrices, D_(IM) and W_(NJ), may beselected to minimize the difference in between the quality of the imageobtained using the transform matrices (i.e. D_(IM)·C_(MN)·W_(NJ)) andthe quality of a down-sampled DCT reference image, V_(IJ), obtainedusing the other down-sampling method. That is, the transform matrices,D_(IM) and W_(NJ), are selected to satisfy the following optimizationproblem:

$\min\limits_{D_{IM},W_{NJ}}\left( {{{{D_{IM} \cdot C_{MN} \cdot W_{NJ}} - V_{IJ}}}^{2} + {\lambda \cdot r_{g}}} \right)$where D_(IM)·C_(MN)·W_(NJ) is a down-sampled image obtained using thetransform matrices, V_(IJ) is the DCT reference image which represents adown-sampled image obtained using another down-sampling method. r_(g) isa measure of the computational complexity associated with using thetransform matrices D_(IM), W_(NJ) to down-sample an image C_(MN) and, λ,is a quality and complexity trade-off parameter.

The quality and complexity trade-off parameter, λ, may be used tobalance the trade-off between the visual quality and computationalcomplexity. That is, the quality complexity trade-off parameter may beused to adjust whether optimization will be more or less focused onvisual quality or complexity.

As used in the above equation, the quality and complexity trade-offparameter, λ, is applied to the computational complexity associated withusing the transform matrices. In this case, a greater quality complexitytrade-off parameter, λ, will result in more importance being placed oncomplexity. Where the quality complexity trade-off parameter, λ, isapplied in this manner, an example of a suitable value may be 0.1.However, it will be appreciated that various values for the quality andcomplexity trade-off parameter, λ, may also be used.

Where a quality and complexity trade-off parameter, λ, is used, themethod may further include a step of retrieving the quality andcomplexity trade-off parameter. This step may be included in the step102 of obtaining the optimization parameters. The quality complexitytrade-off parameter, λ, may be retrieved from a memory. In otherembodiments, the a quality and complexity trade-off parameter, λ, may beobtained via user input.

The computational complexity may be a function of the absolute values ofelements in the row-size transform matrix and the column transformmatrix. That is, in some embodiments, the computational complexity maybe calculated as: r_(g)=|D_(IM)|+|W_(NJ)|, where |·| defines the l₁ normof a matrix.

The computational complexity may, therefore, be a function of how farthe elements of the matrices are from zero. It will be appreciated thatthe computational complexity of the transform matrices will be minimizedwhen all elements of the transform matrices are zero. Of course, it isundesirable to have transform matrices where all elements are zero sinceit would not produce a down-sampled image and the quality, wouldtherefore, be undesirable.

To determine how far the elements of the matrices are from zero, themethod may include a step of calculating the summation of the absolutevalues of elements in the transform matrices.

In some embodiments, the complexity model is established to ignore thenon-zero effect of matrix elements with values greater than a thresholdw_(o), d_(o) on computational complexity. For example, the complexitymodel may be:r _(g) =|D _(IM)|_(|d) _(im) _(|<do) +|W _(NJ)|_(|w) _(nj) _(|<wo)

That is, the method may comprise a step of calculating the summation ofthe absolute value of all elements in the row transform matrix and thecolumn size matrix that are below two thresholds w_(o), d_(o). In manycases, the thresholds may be the same (i.e. w_(o)=d_(o)).

Besides the number of non-zero elements in the row transform matrix andthe column transform matrix, the complexity for computing a down-sampledDCT image using the transform matrices is also related to howmultiplications may be implemented. In general, a multiplication may beapproximated by a series of additions and shifts.

In some embodiments, the computational complexity may be related to thenumber of non-zero elements in the binary representation of the number.That is, it is considered more desirable to have a zero in a binaryrepresentation than a one. Accordingly, the computational complexity mayevaluate the number of non-zero elements in the binary representation ofeach of the row transform matrix and the column transform matrix. Thismay be done by locating a computationally more desirable value withinthe neighborhood of a current element of the transform matrix. Thecomputational complexity of the value may be a function of thedifference between the current element of the transform matrix and thecomputationally more desirable value.

More particularly, for any element, b, in one of the transform matrices,a set of binary digits, {a_(i)}, may be found which satisfies theequation:

$\left\{ a_{i} \right\} = {\arg\;{\min\limits_{{{b - {\sum{({a_{i}2^{- i}})}}}} \leq {{b}n}}{\sum{a_{i}}}}}$where n is a rounding parameter, which is used to adjust how large theneighborhood of an element, b, may be. That is, n is used to adjust howclose the element, b, must be to another value having more zeros in itsbinary representation before the other value will be considereddesirable. The binary representation may be converted back into adecimal number system representation to obtain a quantized decimalvalue. The procedure of locating a neighboring value which is lesscomplex than the current element, b, may be referred to as aquantization procedure, Q(b).

In some embodiments, a binary number system is used in which everybinary digit may be signed. That is, the binary digits may be an elementof the set consisting of positive one (1), zero (0), or minus one (−1).(i.e. α_(i)ε{1, −1, 0)). In some embodiments, such a binaryrepresentation may be used since it permits binary representations witha maximum number of zeros (0). For example, if we consider the binaryrepresentation of the number 0.46875 in a binary system in which eachelement must be selected from the set consisting of only one (1) andzero (0), the binary representation would be 0.01111. That is,0.46875=(0×2⁰)+(0×2⁻¹)+(1×2⁻²)+(1×2⁻³)+(1×2⁻⁴)+(1×2⁻⁵)Using this number system, four ones are needed to represent the number.In contrast, the same number could be represented using the signedsystem mentioned above using only two ones:0.46875=(0×2⁰)+(1×2⁻¹)+(0×2⁻²)+(0×2⁻³)+(0×2⁻⁴)+((−1)×2⁻⁵).

Typically, the magnitude of all elements in the transform matricesD_(IM), W_(NJ) is in the range [0, 8). That is, the magnitudes of allelements is greater than or equal to 0 and less than 8. If thisassumption is made, then the integer portion of the elements may berepresented with three binary digits (i.e. 2², 2¹, 2⁰). The number ofbinary digits used to represent the decimal fraction portion of theelements may depend on the desired visual quality of down-sampled DCTimages obtained using the matrices. For example, if fewer binary digitsare used to represent the decimal fraction portion of the elements, thecomputational complexity associated with using the transform matriceswill be reduced, but the visual quality of the down-sampled DCT imagesmay also be reduced. In some embodiments, fifteen (15) binary digits areallocated for representing the decimal fraction portion of the elementsof the transform matrices.

If the binary system discussed above is used, the quantization procedurefor an element, b, may be represented as:

${{Q(b)} = {\sum\limits_{i = {- 2}}^{i = 15}{a_{i} \cdot 2^{- i}}}},{a_{i} \in \left\{ {1,{- 1},0} \right\}},{{{where}\mspace{14mu}\left\{ a_{i} \right\}} = {\arg\;{\min\limits_{{{b - {\sum{({a_{i}2^{- i}})}}}} \leq {{b}n}}{\sum{a_{i}}}}}}$

That is, the quantization procedure referred to above determines a setof binary numbers, {a_(i)} for a computationally more desirable value.Then the decimal representation of the computationally more desirablevalue is determined.

The difference between the current element, b, of the transform matrixand the computationally more desirable value may then be found. Forexample, the computational complexity associated with the number ofnon-zero elements in the binary representation of the elements of thetransform matrices may be determined as:r _(q) =|D _(IM) −Q(D _(IM))|+|W _(NJ) −Q(W _(NJ))|where Q(D_(IM)), Q(W_(NJ)) mean to apply a quantization procedure toeach element of the row transform matrix and the column transform matrixrespectively.

Since the computational complexity is a function of both the number ofnon-zero decimal representations in the transform matrices and thenumber of non-zero elements in the binary representation of the elementsof the transform matrices, the computational complexity may determinedas:r _(q)=(|D _(IM|) _(|d) _(im) _(|<do) +|W _(NJ|) _(|w) _(nj)_(|<wo))+ρ·(|D _(IM) −Q(D _(IM))|+|W _(NJ) −Q(W _(NJ))|)where ρ is a computational complexity tradeoff parameter used to balancethe relative importance on the computational complexity of the number ofnon-zero elements in the transform matrices and the difference betweencurrent weights in the transform matrix and weights that are close tothe current weights but which would have more zeros in their binaryrepresentation. In some embodiments, ρ is equal to 0.5.

Referring now to FIG. 3, a method of obtaining transform matricesaccording to one aspect of the present application is shown. In thisembodiment, a multiple-layer network is used to find transform matriceswhich minimize the joint cost of the transform matrices. That is, amultiple-layer network is used to find transform matrices which offer adesired balance between computational complexity and visual quality.

At step 302, transform matrices are initialized. Referring to FIGS. 4Aand 4B a multiple-layer network representation of the transform matricesis shown. A three layer network structure is used, in which there is aninput layer 402, representing an input image C_(MN) of size M×N. Thereis also a hidden layer 404, representing an intermediary matrix Y_(IN)of size I×N. The multiple-layer network structure also contains anoutput layer 406, representing a down-sampled image Z_(IJ) of size I×J.Connections 408, 410 are selectively built up among units in each twolayers to simulate the matrix multiplication operation in the lineartransform. That is, connections 408 between the input layer 402 and thehidden layer 404 represent the application of the row transform matrixD_(IM) to the input image C_(MN). Similarly, connections 410 between thehidden layer 404 and the output layer 406 represent the application ofthe column transform matrix W_(NJ) to the intermediary matrix Y_(IN).

Connections between the layers are established according to a number ofrules. With respect to the connections 408 between the input layer andthe hidden layer, connections are established from units in a givencolumn (i.e. n₁) of the input layer to units of the same column (i.e.n₁) in the hidden layer. That is, the input layer and the hidden layerhave the same number of columns. Units in a given column of the inputlayer are fully connected to units in the same column of the hiddenlayer. Valid connections between any of two columns of the input layerand the hidden layer share the same weight matrix i.e. D_(IM).

Similarly, connections 410 between the hidden layer 404 and the outputlayer 406 are built up among rows. Connections are established fromunits in a given column (i.e. i₁) of the hidden layer 404 to units inthe same column of the output layer 406 (i.e. i₁). Valid connections 410between any two rows of the hidden layer and the output layer share thesame weight matrix i.e. W_(NJ).

Accordingly, the output of the hidden layer 404 is computed asY_(IN)=D_(IM)·C_(MN) and the output of the output layer 406 is computedas Z_(IJ)=Y_(IN)·W_(NJ)=D_(IM)·C_(MN)·W_(NJ).

Referring again to FIG. 3, during the step of initialization, an initialstarting weight may be placed in the elements of the transform matricesD_(IM), W_(NJ). The initial starting weight represents a starting pointfrom which a decay procedure will be applied. In some embodiments,random weights may be placed in the elements of the transform matricesD_(IM), W_(NJ).

At step 304, a learning with forgetting algorithm is applied to thetransform matrices to obtain transform matrices with a reduced learningobjective function. This stage is used to find a skeleton structure ofthe transform matrices.

It will be appreciated that due to a random initialization of theconnection weights, some redundant connections may possess an initialweight with a large value. Accordingly, the learning with forgettingstage will apply a constant decay to all elements in order to removeredundant connections which are present due to a random initializationof the weights. Such redundant connections are removed as much aspossible. This approach leads to a constant decay for all non-zeroelements, forcing as many elements to be zero as possible.

The learning objective function at this stage is:J _(f) =∥D _(IM) C _(MN) W _(NJ) −V _(IJ)∥² +λ·r _(g)

where J_(f) is the learning objective function, D_(IM) is the rowtransform matrix, W_(NJ) is the column transform matrix, C_(MN) is theDCT image, D_(IM)·C_(MN)·W_(NJ) is the down-sampled DCT image obtainedusing the transform matrices, V_(IJ) is a down-sampled DCT referenceimage obtained using another down-sampling method, r_(g) is thecomputational complexity associated with using the transform matricesD_(IM), W_(NJ) to down-sample an image C_(MN) and λ is a quality andcomplexity trade-off parameter used to balance a trade-off betweenvisual quality of down-sampled DCT images and the computationalcomplexity of obtaining the down-sampled DCT images. In this case, thecomputational complexity is a function of the absolute value of allelements of the transform matrices. That is,r_(g)=r_(f)=|D_(IM)|+|W_(NJ)|, where |·| defines the l₁ norm of amatrix.

Since the complexity for this stage is a function of the absolute valueof all elements in the matrices, the complexity is minimized if everyelement is zero.

It will also be noted that the visual quality at this stage is measuredas the mean square error of the visual quality of the down-sampled DCTimage produced using the transform matrices and the visual quality ofthe down-sampled DCT image produced by applying another down-samplingmethod to the same initial image.

As illustrated in FIG. 5, the learning with forgetting algorithmincludes a step of adjusting matrices by a learning amount. The learningamount may be calculated by using a back propagation algorithm.

First, the input signal is passed forward to compute network outputs:Y _(IN) D _(IM) ·C _(MN)

Z _(IJ) =Y _(IN) W _(NJ)

A network error is then determined by comparing the network output tothe output for the down-sampled DCT reference image, which was producedusing another down-sampling domain method. The network error is thenpropagated backward.ΔZ _(IJ) =Z _(IJ) −V _(IJ)

(ΔY)_(IN)=(ΔZ)_(IJ)·(W ^(t))_(JN)

The learning amounts may then be determined for each of the transformmatrices. The learning amounts may be determined as:

${\Delta\; D} = {{\frac{1}{2} \cdot \frac{\partial J_{f}}{\partial D}} = {{\left( {\Delta\; Y} \right)_{IN} \cdot \left( C^{t} \right)_{NM}} + {\lambda \cdot {{sgn}\left( D_{IM} \right)}}}}$${\Delta\; W} = {{\frac{1}{2} \cdot \frac{\partial J_{f}}{\partial W}} = {{\left( Y^{t} \right)_{NI} \cdot \left( {\Delta\; Z} \right)_{NI} \cdot \left( {\Delta\; Z} \right)_{IJ}} + {\lambda \cdot {{sgn}\left( W_{NJ} \right)}}}}$where sgn(x) is the sign function as:

${{sgn}(x)} = \left\{ \begin{matrix}{\frac{1}{2},{x > 0}} \\{0,{x = 0}} \\{{- \frac{1}{2}},{x < 0}}\end{matrix} \right.$

The learning amounts may then be applied to the transform matrices toobtain adjusted transform matrices:D _(IM) ^((n+1)) =D _(IM) ^((n)) −α·ΔDW _(NJ) ^((n+1)) =W _(NJ) ^((n)) −α·ΔWwhere the superscripts (n) and (n+1) accord to the nth and (n+1)thiterations and α is a small positive number named the learning factor orlearning step size parameter. The learning factor is used to adjust thestep size of learning operations. That is, it is used to adjust howgreat the adjustment will be with each iteration of learning. In someembodiments, the learning factor is 1×10⁻⁶.

Typically, there will be multiple learning with forgetting iterations.In one embodiment, at step 504, a determination is made regardingwhether the decrement of the learning objective function was less than apredetermined threshold. If it was not, another iteration of learningwith forgetting will be performed. If it was less than the predeterminedthreshold, then the learning with forgetting stage will be considered tobe completed. That is, the learning with forgetting stage is completewhen each iteration of learning no longer yields a sufficientimprovement to the transform matrices in terms of minimizing the jointcost.

In some embodiments, the predetermined threshold is zero. That is, thelearning objective function will be completed when the joint cost of thetransform matrices is no longer improving with each iteration.

Learning with forgetting normally ends with a skeleton structure but alarge distortion of ∥D·C·W−V∥². That is, a constant decay to elementswith large values will introduce a large distortion to the visualquality, measured as ∥x_(I J)−T⁻¹(D_(IM)·C_(MN)·W_(NJ))∥², where x_(ij)is a spatial domain down-sampled image and T⁻¹(x) is an inverse DCTtransform.

As illustrated in FIG. 3, a step 306 of performing a selective learningwith forgetting algorithm may then be used to tune the structure forbetter trade-off between the distortion of ∥D·C·W−V∥² and thecomplexity. The selective learning with forgetting stage protectscertain large elements from decay so that they can be trained to focuson providing better visual quality.

Since the selective learning with forgetting stage protects certainlarge elements from decay, the complexity model may include at least oneselective learning with forgetting threshold parameter (w_(o) or d_(o))to be used for determining what is or is not considered a large element.In some embodiments, each of the transform matrices has its ownselective learning with forgetting threshold parameter (w_(o) or d_(o)).In other embodiments, the selective learning with forgetting thresholdparameter (w_(o) or d_(o)) is the same for both transform matrices.

The selective learning with forgetting threshold parameter (w_(o) ord_(o)) has the effect of adjusting the trade-off between visual qualityand complexity since a higher threshold may result in a higher qualityimage, but will result in less elements of the transform matrices whichhave been pushed toward zero. In some embodiments, the selectivelearning with forgetting threshold parameter (w_(o) or d_(o)) is 0.1.

The learning objective function at this stage is very similar to thelearning objective at the learning with forgetting stage:J _(f) =∥D _(IM) C _(MN) W _(NJ) −V _(IJ)∥² +λ·r _(g)

where J_(f) is the learning objective function, D_(IM) is the rowtransform matrix, W_(NJ) is the column transform matrix, C_(MN) is theDCT image, D_(IM)·C_(MN)·W_(NJ) is the down-sampled DCT image obtainedusing the transform matrices, V_(IJ) is a down-sampled DCT referenceimage obtained using another down-sampling method, r_(g) is selectivelearning with forgetting computational complexity associated with usingthe transform matrices D_(IM), W_(NJ) to down-sample an image C_(MN) andλ is a quality and complexity trade-off parameter used to balance atrade-off between visual quality of down-sampled DCT images and thecomputational complexity of obtaining the down-sampled DCT images.

In this case, however, the complexity model is established to ignore thenon-zero effect of matrix elements with values greater than thethreshold on the complexity. For example, the complexity model for thelearning with forgetting stage may be:r _(g) =r _(q) =|D _(IM)|_(|d) _(im) _(|<do) +|W _(NJ)|_(|w) _(nj)_(|<wo)

That is, the method may comprise a step of calculating the summation ofthe absolute value of all elements in the row transform matrix and thecolumn size matrix that are below two thresholds w_(o), d_(o). In manycases, the thresholds may be the same (i.e. w_(o)=d_(o)).

As discussed previously, besides the number of non-zero elements in therow transform matrix and the column transform matrix, the complexity forcomputing a down-sampled DCT image using the transform matrices is alsorelated to how multiplications may be implemented. The computationalcomplexity may be related to the number of non-zero elements in thebinary representation of the number. That is, it is considered moredesirable to have a zero in a binary representation than a one.Accordingly, the computational complexity may evaluate the number ofnon-zero elements in the binary representation of each of the rowtransform matrix and the column transform matrix using the algorithmdiscussed above. This may be done by locating a computationally moredesirable value within the neighborhood of a current element of thetransform matrix. The computational complexity of the value may be afunction of the difference between the current element of the transformmatrix and the computationally more desirable value.

More particularly, for any element, b, in one of the transform matrices,a set of binary digits, {a_(i)}, may be found which satisfies theequation:

$\left\{ a_{i} \right\} = {\arg\;{\min\limits_{{{b - {\sum{({a_{i}2^{- i}})}}}} \leq {{b}n}}{\sum{a_{i}}}}}$where n is a rounding parameter, which is used to adjust how large theneighborhood of an element, b, may be. That is, n is used to adjust howclose the element, b, must be to another value having more zeros in itsbinary representation before the other value will be considereddesirable. The binary representation may be converted back into adecimal number system representation to obtain a quantized decimalvalue. The procedure of locating a neighboring value which is lesscomplex than the current element, b, may be referred to as aquantization procedure, Q(b).

If the binary system discussed above is used, the quantization procedurefor an element, b, may be represented as:

${{Q(b)} = {\sum\limits_{i = {- 2}}^{i = 15}{a_{i} \cdot 2^{- i}}}},{a_{i} \in \left\{ {1,{- 1},0} \right)},{{{where}\mspace{14mu}\left\{ a_{i} \right\}} = {\arg\;{\min\limits_{{{b - {\sum{({a_{i}2^{- i}})}}}} \leq {{b}n}}{\sum{a_{i}}}}}}$

That is, the quantization procedure referred to above determines a setof binary numbers, {a_(i)} for a computationally more desirable valuethat is within the neighborhood of the element, b. Then the decimalrepresentation of the computationally more desirable value isdetermined.

The difference between the current element, b, of the transform matrixand the computationally more desirable value may then be found. Forexample, the computational complexity associated with the number ofnon-zero elements in the binary representation of the elements of thetransform matrices may be determined as:r _(q) =|D _(IM) −Q(D _(IM))|+|W _(NJ) −Q(W _(NJ))|where Q(D_(IM)), Q(W_(NJ)) mean to apply a quantization procedure toeach element of the row transform matrix and the column transform matrixrespectively.

Since the computational complexity is a function of both the number ofnon-zero decimal representations in the transform matrices and thenumber of non-zero elements in the binary representation of the elementsof the transform matrices, the computational complexity in the selectivelearning with forgetting algorithm may determined as:r _(q)=(|D _(IM)|_(|d) _(im) _(|<do) +|W _(NJ)|_(|w) _(nj)_(|<wo))+ρ·(|D _(IM) −Q(D _(IM))|+|W _(NJ) −Q(W _(NJ))|)where ρ is a computational complexity tradeoff parameter used to balancethe relative importance on the computational complexity of the number ofnon-zero elements in the transform matrices and the difference betweencurrent weights in the transform matrix and weights that are close tothe current weights but which would have more zeros in their binaryrepresentation. In some embodiments, ρ is equal to 0.5.

The quantization procedure discussed above, which examines whether thereis a neighboring value which is computationally less complex, isgenerally applied at the selective learning with forgetting stagebecause its corresponding contribution to the complexity function is ata level similar to r_(q)=|D_(IM)|_(|d) _(im) _(|<do)+|W_(NJ)|_(|w) _(nj)_(|<wo) which is much less than r_(f)=|D_(IM)|+|W_(NJ)|.

Note, however, that regardless of which stage the quantization procedureis applied at, the computational complexity tradeoff parameter ρ may beused to balance the relative importance on the computational complexityof the number of non-zero elements in the transform matrices and thedifference between current weights in the transform matrix and weightsthat are close to the current weights but which would have more zeros intheir binary representation.

Referring now to FIG. 6, a method of applying a selective learning withforgetting algorithm to the transform matrices is shown. At step 602,the transform matrices are adjusted by a learning amount.

The learning amount is determined in much the same way as the learningamount is determined for the learning with forgetting stage.

First, the input signal is passed forward to compute network outputs:Y _(IN) =D _(IM) ·C _(MN)

Z _(IJ) =Y _(IN) ·W _(NJ)

A network error is then determined by comparing the network output tothe output for the down-sampled DCT reference image, which was producedusing another down-sampling domain method. The network error is thenpropagated backward.ΔZ _(IJ) =Z _(IJ) −V _(IJ)

(ΔY)_(IN)=(ΔZ)_(IJ)·(W ^(t))_(JN)

The learning amounts may then be determined for each of the transformmatrices. The formula for determining the learning amounts in theselective learning with forgetting stage differs from the formula fordetermining the learning amounts for the learning with forgetting stage.In the selective learning with forgetting stage, the learning amountsmay be determined as:ΔD=(ΔY)_(IN)·(C ^(t))_(NM) +λ·thr(D _(IM) ,d _(o))+λ·ρ·sgn(D _(IM) −Q(D_(IM)))ΔW=(Y ^(t))_(NI)·(ΔZ)_(IJ) +λ·thr(W _(NJ) ,w _(o))+λ·ρ·sgn(W _(NJ) −Q(W_(NJ)))where:

${{thr}\left( {x,\theta} \right)} = \left\{ {{\begin{matrix}{\frac{1}{2},{\theta > x > 0}} \\{0,{{x = 0};{x \geq \theta};{x \leq {- \theta}}}} \\{{- \frac{1}{2}},{{- \theta} < x < 0}}\end{matrix}{{sgn}(x)}} = \left\{ \begin{matrix}{\frac{1}{2},{x > 0}} \\{0,{x = 0}} \\{{- \frac{1}{2}},{x < 0}}\end{matrix} \right.} \right.$and λ is the quality and complexity trade-off parameter and ρ is thecomputational complexity tradeoff parameter.

As before with the learning with forgetting algorithm, the learningamounts may then be applied to the transform matrices to obtain adjustedtransform matrices:D _(IM) ^((n+1)) =D _(IM) ^((n)) −α·ΔDW _(NJ) ^((n+1)) =W _(NJ) ^((n)) −α·ΔWwhere the superscripts (n) and (n+1) accord to the nth and (n+1)thiterations and α is a small positive number named the learning factor orlearning step size parameter. As before, the learning factor is used toadjust the step size of learning operations. The learning factor for theselective learning with forgetting stage may be a different value thanthe learning factor for the learning with forgetting stage.

Typically, there will be multiple selective learning with forgettingiterations. In one embodiment, at step 604, a determination is maderegarding whether the decrement of the learning objective function wasless than a predetermined threshold. If it was not, another iteration oflearning with forgetting will be performed. If it was less than thepredetermined threshold, then the selective learning with forgettingstage will be considered to be completed. That is, the selectivelearning with forgetting stage is complete when each iteration oflearning no longer yields a sufficient improvement to the transformmatrices in terms of minimizing the joint cost.

In some embodiments, the predetermined threshold is zero. That is, thelearning objective function will be completed when the joint cost of thetransform matrices is no longer improving with each iteration.

Following the selective learning with forgetting algorithm, in someembodiments the computational complexity of the transform matrices maybe further reduced by combining arithmetic operations.

Referring now to FIG. 7, according to one aspect of this application, amethod is provided for down-sampling a DCT image of size M×N in the DCTdomain to produce a down-sampled image of size I×J.

In step 702, transform matrices are obtained. The transform matricesinclude a row transform matrix of size I×M and a column transform matrixof size N×J. The transform matrices may have been produced according toany of the methods of producing transform matrices for down-sampling aDCT image directly in the DCT domain outlined above. That is, the rowtransform matrix and the column transform matrix are matrices that havebeen selected to jointly optimize the visual quality of down-sampled DCTimages obtained using the matrices and minimize the computationalcomplexity associated with applying the matrices to DCT images. Thevisual quality of the down-sampled images is determined with referenceto the visual quality of another down-sampled image obtained usinganother down-sampling method. In some embodiments, the matrices havebeen determined using a multiple-layer network to solve an optimizationproblem.

In some embodiments, the step 702 of obtaining includes looking up therow transform matrix and the column transform matrix in a memory. Inother embodiments, the step 702 of obtaining includes looking upappropriate transform matrices in a database of transform matrices. Insuch embodiments, the database may include transform matrices forvarious down-sampling ratios and the step of obtaining includesselecting the transform matrices with a desired down-samplingcharacteristic, such as a desired down-sampling ratio.

In some embodiments, the database may be stored remotely on a server.The step of obtaining includes a step of requesting appropriatetransform matrices from the server and receiving the row transformmatrix and column transform matrix from the server. To ensure that thecorrect transform matrices are obtained, the step of requesting mayinclude transmitting system identification data to the server. Forexample, the identification data may include data identifying a displayresolution of a device which will be used to display the down-sampledimage and/or data identifying processor speed information of a processorwhich will be used to transform the DCT image into the down-sampled DCTimage.

In this way, transform matrices may be received which are best suitedfor the device's display or processing capabilities.

Following the step of obtaining, a step 704 of determining anintermediary matrix as the product of the DCT image and one of eitherthe row transform matrix or the column transform matrix. Since matrixmultiplication is not commutative, the arrangement of multiplication maybe important. The row-transform matrix, D_(IM), is generally to the leftof the DCT image, C_(MN), in the transformation formula and thecolumn-transform image, W_(NJ), is generally to the right of the DCTimage in the transformation formula.

That is, since D_(IM)·C_(MN)·W_(NJ)≠W_(NJ)·C_(MN)·D_(IM), the latterexpression is not used. However, since matrix multiplication isassociative, the order may be modified to some extent. The row sizematrix may be multiplied with the DCT image and the result multipliedwith the column transform matrix (i.e. (D_(IM)·C_(MN))·W_(NJ)), oralternatively the DCT image may be multiplied with the column transformmatrix and the result multiplied with the row transform matrix (i.e.D_(IM)·(C_(MN)·W_(NJ))).

Accordingly, in step 704 an intermediary matrix, K, is obtained as theproduct of the DCT image and one of either the row transform matrix orthe column transform matrix is obtained as either: K=(D_(IM)·C_(MN)) orK=(C_(MN)·W_(NJ))

Next, at step 706, the down-sampled DCT image is obtained as the productof the intermediary matrix K and the transform matrix that is not usedin the previous step of determining (either D_(IM) or W_(NJ)). Forexample, if the intermediary matrix K is determined asK=(D_(IM)·C_(MN)), then the down-sampled DCT image is obtained asX_(IJ)=(K·W_(NJ)). Alternatively, if the intermediary matrix K isdetermined as K=(C_(MN)·W_(NJ)), then the down-sampled DCT image isobtained as X_(IJ)=(D_(IM)·K).

It will be appreciated that transcoders for down-sampling images areused in wide variety of electrical devices. Accordingly, the methodsdescribed above may be implemented on any suitable electronic devicecomprised of a processor and a memory.

By way of example, in one embodiment, illustrated in FIG. 8, the methodsdescribed above are used to down-sample images to a size suitable fordisplay on a display of a mobile device 802. The mobile device 802 isconnected to a server 804. The mobile device 802 may be connected to theserver 804 in any one of a number of methods. For example, the mobiledevice 802 may be connected to the server through a cellular network, awireless wide area network (WWAN), or a wireless local area network(WLAN). It will be appreciated that other methods of connection, bothwired and wireless, may also be used.

The server 804 may be an Internet gateway which provides access to theInternet. In some embodiments, the server 804 includes a transcoder 808which is configured to down-sample images from a size M×N DCT image to asize I×J DCT image. In some embodiments, the M×N DCT images are imagesreceived from the Internet.

The transcoder 808 comprises a processor 810 and a memory 812. Theprocessor 810 may be configured to execute the steps of any of thepreceding methods to find transform matrices for down-sampling a DCTimage directly in the DCT domain. The processor 810 may also beconfigured to down-sample a DCT image directly in the DCT domain usingthe transform matrices. In one embodiment, the memory 812 is used tostore transform matrices. The memory 812 may also be used to store anyof the optimization parameters.

The processor 810 is connected to the memory 812. In one embodiment, theprocessor is configured to select a spatial domain down-sampling methodand to apply the selected spatial domain down-sampling method to a DCTimage to produce a DCT reference image. The processor 810 may beconfigured to store optimization parameters which have been found tominimize or reduce the optimization problem to the memory 812.

The processor 810 may be configured to determine or further optimizetransform matrices during periods when the processor is experiencinglittle use.

In some embodiments, the processor 810 is configured to produce adown-sampled image of size I×J according to the method illustrated inFIG. 7 and described above.

In some embodiments, the transcoder is included on the mobile device802. The mobile device 802 also has a display for displaying images anda communication module for communicating with the server. The mobiledevice 802 may also include a processor connected to the communicationmodule. The processor is configured to execute the down-sampling methodillustrated in FIG. 7. That is, the processor is configured to obtainfrom the server the transform matrices and to determine an intermediarymatrix as the product of the DCT image and one of either the rowtransform matrix or the column transform matrix and to determine adown-sampled DCT image of size I×J as the product of the intermediarymatrix and the transform matrix which was not used in determining theintermediary matrix. This embodiment, in which the server determines andprovides the transform matrices to the mobile device 802 which thenperforms multiplication operations using the transform matrices may beparticularly useful for down-sampling images which are locally producedon the mobile device using, for example, a camera. This configurationpermits a user to down-sample a locally produced image obtained usingthe camera to minimize the bandwidth used in transmitting the image overthe network. Accordingly, in some embodiments, the mobile electronicdevice 802 includes a camera.

Accordingly, in one aspect the present application provides a method ofdown-sampling a DCT image of size M×N in the DCT domain to produce adown-sampled image of size I×J comprising steps of: a) obtainingtransform matrices comprising a row transform matrix of size I×M and acolumn transform matrix of size N×J; b) determining an intermediarymatrix as the product of the DCT image and one of either the rowtransform matrix or the column transform matrix; and c) determining thedown-sampled DCT image as the product of the intermediary matrix and thetransform matrix not used in the previous step of determining.

In another aspect, the present application provides a transcoder fortranscoding a DCT image. The transcoder has a memory having transformmatrices stored thereon. The transform matrices comprise a row transformmatrix of size I×M and a column transform matrix of size N×J. Thetranscoder also includes a processor connected to the memory andconfigured to obtain from the memory the transform matrices, and todetermine an intermediary matrix as the product of the DCT image and oneof either the row transform matrix or the column transform matrix. Theprocessor is also configured to determine a down-sampled DCT image ofsize I×J as the product of the intermediary matrix and the transformmatrix not used in determining intermediary matrix.

In another aspect, the present application provides a mobile electronicdevice having a transcoder for down-sampling a DCT image. The mobileelectronic device includes a display for displaying images. The mobileelectronic device includes a communication module for communicating witha server and a processor connected to the communication module. Theprocessor is configured to obtain from the server the transform matricesincluding a row transform matrix and a column transform matrix. Theprocessor is also configured to determine an intermediary matrix as theproduct of the DCT image and one of either the row transform matrix orthe column transform matrix. The processor is also configured todetermine a down-sampled DCT image of size I×J as the product of theintermediary matrix and the transform matrix not used in determining theintermediary matrix.

In yet a further aspect, the present application provides a method ofselecting transform matrices for use in a transcoder for down-sampling aDCT image directly in the DCT domain. The method comprises: a) obtaininga training set by applying a pre-selected down-sampling method to a DCTreference image to obtain a down-sampled DCT reference image; b)initializing the transform matrices; and c) applying a learning withforgetting algorithm to the transform matrices to obtain transformmatrices which reduce a learning objective function, wherein thelearning objective function is a function of the visual quality of adown-sampled DCT image obtained using the transform matrices relative tothe visual quality of the down-sampled DCT reference image, and thelearning objective function is also a function of the computationalcomplexity of applying the transform matrices.

In yet another aspect, the present application provides an electronicdevice having a memory for storing transform matrices and a trainingset. The transform matrices include a row transform matrix and a columntransform matrix. The training set includes at least one DCT referenceimage and a corresponding down-sampled DCT reference image obtainedusing a spatial domain down-sampling method. The electronic device alsoincludes a processor connected to the memory for obtaining transformmatrices for down-sampling a DCT image directly in a DCT domain. Theprocessor is configured to initialize the transform matrices and toapply a learning with forgetting algorithm to the transform matrices toobtain transform matrices which reduce a learning objective function.The learning objective function is a function of the visual quality of adown-sampled DCT image obtained using the transform matrices relative tothe visual quality of the down-sampled DCT reference image. The learningobjective function is also a function of the computational complexityassociated with down-sampling using the transform matrices.

In another aspect, the present application provides a method ofselecting transform matrices for use in a transcoder for down-sampling aDCT image of size M×N to a down-sampled DCT image of size I×J directlyin a DCT domain. The method comprises: a) selecting a spatial domaindown-sampling method; b) applying the selected spatial domaindown-sampling method to the DCT image to produce a down-sampled DCTreference image; and c) selecting transform matrices that minimize anoptimization problem, wherein the optimization problem is a function ofthe error between down-sampled image obtained using the transformmatrices and the down-sampled DCT reference image, and the optimizationproblem is also a function of the computational complexity of applyingthe transform matrices, the transform matrices comprising a rowtransform matrix and a column transform matrix.

In another aspect, the present application provides an electronicdevice. The electronic device includes a memory for storing transformmatrices. The transform matrices include a row transform matrix and acolumn transform matrix. The electronic device further includes aprocessor connected to the memory for obtaining transform matrices fordown-sampling a DCT image of size M×N to a down-sampled DCT image ofsize I×J directly in a DCT domain. The processor is configured to selecta spatial domain down-sampling method and to apply the selected spatialdomain down-sampling method to the DCT image to produce a DCT referenceimage. The processor is also configured to store transform matrices tothe memory which minimize an optimization problem. The optimizationproblem is a function of the error between down-sampled image obtainedusing the transform matrices and the DCT reference image. Theoptimization problem is also a function of the computational complexityof applying the transform matrices to down-sample the DCT image.

Other aspects and features of the present application will be apparentto those of ordinary skill in the art from a review of the followingabove description when considered in conjunction with the drawings.

Certain adaptations and modifications of the invention will be obviousto those skilled in the art when considered in light of thisdescription. Therefore, the above discussed embodiments are consideredto be illustrative and not restrictive, the scope of the invention beingindicated by the appended claims rather than the foregoing description,and all changes which come within the meaning and range of equivalencyof the claims are therefore intended to be embraced therein.

What is claimed is the following:
 1. A method of selecting transformmatrices for down-sampling in the discrete cosine transform (DCT)domain, the method comprising: initializing the transform matrices;obtaining at least one optimization parameter; and selecting transformmatrices to minimize an optimization expression, the optimizationexpression is a function of the at least one optimization parameter,wherein the at least one optimization parameter comprises a complexitytrade-off parameter to balance the relative importance of a plurality offactors on the computational complexity of selecting transform matrices.2. The method of claim 1, wherein the at least one optimizationparameter comprises a quality-and-complexity trade-off parameter tobalance the relative importance of quality of an image down-sampledusing the transform matrices and the computational complexity ofselecting the transform matrices.
 3. The method of claim 1 furthercomprising: converting a DCT reference image into a spatial domainreference image; applying a spatial domain down-sampling method to thespatial domain reference image to produce a spatial domain down-sampledreference image; and converting the spatial domain down-sampledreference image to the DCT domain to produce a down-sampled DCTreference image, wherein the at least one optimization parameterscomprises a quality-and-complexity trade-off parameter to adjust thetrade-off between the error between a down-sampled image obtained usingthe transform matrices and the down-sampled DCT reference image.
 4. Themethod of claim 3, wherein selecting transform matrices includesdetermining the visual quality of the down-sampled DCT image obtained byapplying the transform matrices to the DCT reference image relative tothe visual quality of the down-sampled DCT reference image bycalculating the mean squared error between the down-sampled DCT imageobtained using the transform matrices and the down-sampled DCT referenceimage.
 5. The method of claim 3, wherein the transform matrices comprisea row-transform matrix and a column-transform matrix, and whereinselecting transform matrices includes determining the visual quality ofdown-sampled DCT images obtained by applying the row-transform matrixand the column-transform matrix to the DCT reference image, anddetermining the computational complexity associated with applying therow-transform matrix and the column-transform matrix to down-sample theDCT reference image.
 6. The method of claim 3, wherein the transformmatrices comprise a row-transform matrix and a column-transform matrix,and wherein selecting transform matrices includes selecting transformmatrices that satisfy the optimization expression:${\min\limits_{D_{IM},W_{NJ}}\left( {{{{D_{IM} \cdot C_{MN} \cdot W_{NJ}} - V_{IJ}}}^{2} + {\lambda \cdot r_{g}}} \right)},$where D_(IM) is the row-transform matrix; W_(NJ) is the column-transformmatrix; C_(MN) is the DCT image; D_(IM)·C_(MN)·W_(NJ) is thedown-sampled DCT image obtained using the transform matrices; V_(IJ) isthe down-sampled DCT reference image; r_(g) is a measure of thecomputational complexity associated with using the transform matricesD_(IM), W_(NJ) to down-sample an image C_(MN); and λ is thequality-and-complexity trade-off parameter for balancing the trade-offbetween visual quality of the down-sampled DCT image and thecomputational complexity of obtaining the down-sampled DCT image.
 7. Themethod of claim 1, wherein the computational complexity is a function ofthe absolute values of elements in the transform matrices.
 8. The methodof claim 7, further comprising calculating the summation of the absolutevalue of all elements in the transform matrices.
 9. The method of claim1, wherein the at least one optimization parameter comprises a selectivelearning with forgetting threshold parameter and wherein thecomputational complexity is a function of the absolute values of allelements in the transform matrices below the selective learning withforgetting threshold parameter.
 10. The method of claim 9 furthercomprising calculating the summation of the absolute values of allelements in the transform matrices below the selective learning withforgetting threshold parameter.
 11. The method of claim 1, wherein thecomputational complexity is a function of the number of non-zeroelements in a binary representation of each element of the transformmatrices.
 12. An electronic device comprising: a memory; and a processorconnected to the memory for obtaining transform matrices fordown-sampling in the discrete cosine transform (DCT) domain, theprocessor being configured to: initialize the transform matrices; obtainat least one optimization parameter; and select transform matrices tominimize an optimization expression, the optimization expression is afunction of the at least one optimization parameter, wherein the atleast one optimization parameter comprises a complexity trade-offparameter to balance the relative importance of a plurality of factorson the computational complexity.
 13. The electronic device of claim 12,wherein the at least one optimization parameter is retrieved frommemory.
 14. The electronic device of claim 12, wherein the at least oneoptimization parameter comprises a quality-and-complexity trade-offparameter to balance the relative importance of quality of an imagedown-sampled using the transform matrices and the computationalcomplexity of selecting the transform matrices.
 15. The electronicdevice of claim 12 wherein the processor is further configured to:convert a DCT reference image into a spatial domain reference image;apply a spatial domain down-sampling method to the spatial domainreference image to produce a spatial domain down-sampled referenceimage; and convert the spatial domain down-sampled reference image tothe DCT domain to produce a down-sampled DCT reference image, whereinthe at least one optimization parameter comprises a quality andcomplexity trade-off parameter to adjust the trade-off between the errorbetween a down-sampled image obtained using the transform matrices andthe down-sampled DCT reference image.
 16. The electronic device of claim15, wherein selecting transform matrices includes determining the visualquality of the down-sampled DCT image obtained by applying the transformmatrices to the DCT reference image relative to the visual quality ofthe down-sampled DCT reference image by calculating the mean squarederror between the down-sampled DCT image obtained using the transformmatrices and the down-sampled DCT reference image.
 17. The electronicdevice of claim 15 wherein the transform matrices comprise arow-transform matrix and a column-transform matrix and wherein theoptimization expression is:${\min\limits_{D_{IM},W_{NJ}}\left( {{{{D_{IM} \cdot C_{MN} \cdot W_{NJ}} - V_{IJ}}}^{2} + {\lambda \cdot r_{g}}} \right)},$where D_(IM) is the row-transform matrix; W_(NJ) is the column-transformmatrix; C_(MN) is the DCT image; D_(IM)·C_(MN)·W_(NJ) is thedown-sampled DCT image obtained using the transform matrices; V_(IJ) isthe down-sampled DCT reference image; r_(g) is a measure of thecomputational complexity associated with using the transform matricesD_(IM), W_(NJ) to down-sample an image C_(MN); and λ is thequality-and-complexity trade-off parameter for balancing the trade-offbetween visual quality of the down-sampled DCT image and thecomputational complexity of obtaining the down-sampled DCT image. 18.The electronic device of claim 12, wherein the at least one optimizationparameter comprises a predetermined threshold parameter and wherein theprocessor is configured to ignore the computational complexity due tovalues in the transform matrices that are above the predeterminedthreshold parameter.